IUMJ

Title: Moebius transformations and the Poincare distance in the quaternionic setting

Authors: Cinzia Bisi and Graziano Gentili

Issue: Volume 58 (2009), Issue 6, 2729-2764

Abstract:

In the space $\mathbb{H}$ of quaternions, we investigate the natural, invariant geometry of the open, unit disc $\Delta_{\mathbb{H}}$ and of the open half-space $\mathbb{H}^{+}$. These two domains are diffeomorphic via a Cayley-type transformation. We first study the geometrical structure of the groups of M\"obius transformations of $\Delta_{\mathbb{H}}$ and $\mathbb{H}^{+}$ and identify original ways of representing them in terms of two (isomorphic) groups of matrices with quaternionic entries. We then define the cross-ratio of four quaternions, prove that, when real, it is invariant under the action of the M\"obius transformations, and use it to define the analog of the Poincar\'e distances and differential metrics on $\Delta_{\mathbb{H}}$ and $\mathbb{H} ^{+}$.